On Unconstrained and Constrained Stationary Points of the Implicit Lagrangian

Mangasarian and Solodov (Ref. 1) proposed to solve nonlinear complementarity problems by seeking the unconstrained global minima of a new merit function, which they called implicit Lagrangian. A crucial point in such an approach is to determine conditions which guarantee that every unconstrained stationary point of the implicit Lagrangian is a global solution, since standard unconstrained minimization techniques are only able to locate stationary points. Some authors partially answered this question by giving sufficient conditions which guarantee this key property. In this paper, we settle the issue by giving a necessary and sufficient condition for a stationary point of the implicit Lagrangian to be a global solution and, hence, a solution of the nonlinear complementarity problem. We show that this new condition easily allows us to recover all previous results and to establish new sufficient conditions. We then consider a constrained reformulation based on the implicit Lagrangian in which nonnegative constraints on the variables are added to the original unconstrained reformulation. This is motivated by the fact that often, in applications, the function which defines the complementarity problem is defined only on the nonnegative orthant. We consider the KKT-points of this new reformulation and show that the same necessary and sufficient condition which guarantees, in the unconstrained case, that every unconstrained stationary point is a global solution, also guarantees that every KKT-point of the new problem is a global solution.     

Necessary conditions for optimal control problems with bounded state by a penalty method

Necessary conditions are derived for a general relaxed control problem with unilateral state constraint. The results are also valid for ordinary controls that are solutions of the relaxed problem.A penalty is imposed to change the constrained problem into a sequence of unconstrained problems. The assumptions are on the data of the problem and do not requirea priori verification of hypotheses involving the optimal solution.     

Application of the hypodifferential descent method to the problem of constructing an optimal control

In this paper the problem of optimal control of a nonlinear ODE system with given boundary conditions and the integral restriction on control is considered. With the help of the theory of exact penalty functions the original problem is reduced to the problem of unconstrained minimization of a nonsmooth functional. The necessary minimum conditions in terms of hypodifferentials are found. A class of problems for which these conditions are also sufficient is distinguished. On the basis of these conditions the hypodifferential descent method is applied to the considered problem. Under some additional assumptions the hypodifferential descent method converges in a certain sense.     

Pontryagin Principle for State-Constrained Control Problems Governed by a First-Order PDE System

This paper considers multidimensional control problems governed by a first-order PDE system and state constraints. After performing the standard Young measure relaxation, we are able to prove the Pontryagin principle by means of an -maximum principle. Generalizing the common setting of one-dimensional control theory, we model piecewise-continuous weak derivatives as functions of the first Baire class and obtain regular measures as corresponding multipliers. In a number of corollaries, we derive necessary optimality conditions for local minimizers of the state-constrained problem as well as for global and local minimizers of the unconstrained problem.     

The exact penalty principle

The exact penalty approach aims at replacing a constrained optimization problem by an equivalent unconstrained optimization problem. Most results in the literature of exact penalization are mainly concerned with finding conditions under which a solution of the constrained optimization problem is a solution of an unconstrained penalized optimization problem, and the reverse property is rarely studied. In this paper, we study the reverse property. We give the conditions under which the original constrained (single and/or multiobjective) optimization problem and the unconstrained exact penalized problem are exactly equivalent. The main conditions to ensure the exact penalty principle for optimization problems include the global and local error bound conditions. By using variational analysis, these conditions may be characterized by using generalized differentiation.     

Lipschitzianity of optimal trajectories for the Bolza optimal control problem

In this paper we investigate Lipschitz continuity of optimal solutions for the Bolza optimal control problem under Tonelli’s type growth condition. Such regularity being a consequence of normal necessary conditions for optimality, we propose new sufficient conditions for normality of state-constrained nonsmooth maximum principles for absolutely continuous optimal trajectories. Furthermore we show that for unconstrained problems any minimizing sequence of controls can be slightly modified to get a new minimizing sequence with nice boundedness properties. Finally, we provide a sufficient condition for Lipschitzianity of optimal trajectories for Bolza optimal control problems with end point constraints and extend a result from (J. Math. Anal. Appl. 143, 301–316, 1989) on Lipschitzianity of minimizers for a classical problem of the calculus of variations with discontinuous Lagrangian to the nonautonomous case.     

Extremum conditions for a nonsmooth function in terms of exhausters and coexhausters

The notions of upper and lower exhausters and coexhausters are discussed and necessary conditions for an unconstrained extremum of a nonsmooth function are derived. The necessary conditions for a minimum are formulated in terms of an upper exhauster (coexhauster) and the necessary conditions for a maximum are formulated in terms of a lower exhauster (coexhauster). This involves the problem of transforming an upper exhauster (coexhauster) into a lower exhauster (coexhauster) and vice versa. The transformation is carried out by means of a conversion operation (converter). Second-order approximations obtained with the help of second-order (upper and lower) coexhausters are considered. It is shown how a secondorder upper coexhauster can be converted into a lower coexhauster and vice versa. This problem is reduced to using a first-order conversion operator but in a space of a higher dimension. The obtained result allows one to construct second-order methods for the optimization of nonsmooth functions (Newton-type methods).     

Algorithms for Unconstrained Optimization Problems via Control Theory

Existing algorithms for solving unconstrained optimization problems are generally only optimal in the short term. It is desirable to have algorithms which are long-term optimal. To achieve this, the problem of computing the minimum point of an unconstrained function is formulated as a sequence of optimal control problems. Some qualitative results are obtained from the optimal control analysis. These qualitative results are then used to construct a theoretical iterative method and a new continuous-time method for computing the minimum point of a nonlinear unconstrained function. New iterative algorithms which approximate the theoretical iterative method and the proposed continuous-time method are then established. For convergence analysis, it is useful to note that the numerical solution of an unconstrained optimization problem is none other than an inverse Lyapunov function problem. Convergence conditions for the proposed continuous-time method and iterative algorithms are established by using the Lyapunov function theorem.     

Equivalence of variational inequality problems to unconstrained minimization

In this paper we propose a class of merit functions for variational inequality problems (VI). Through these merit functions, the variational inequality problem is cast as unconstrained minimization problem. We estimate the growth rate of these merit functions and give conditions under which the stationary points of these functions are the solutions of VI. This work was supported by the state key project “Scientific and Engineering Computing”.     

An accelerated multiplier method for nonlinear programming

This paper describes an accelerated multiplier method for solving the general nonlinear programming problem. The algorithm poses a sequence of unconstrained optimization problems. The unconstrained problems are solved using a rank-one recursive algorithm described in an earlier paper. Multiplier estimates are obtained by minimizing the error in the Kuhn-Tucker conditions using a quadratic programming algorithm. The convergence of the sequence of unconstrained problems is accelerated by using a Newton-Raphson extrapolation process. The numerical effectiveness of the algorithm is demonstrated on a relatively large set of test problems.This work was supported by the US Air Force under Contract No. F04701-74-C-0075.     

An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem

In this paper, we consider a class of optimal control problems subject to equality terminal state constraints and continuous state and control inequality constraints. By using the control parametrization technique and a time scaling transformation, the constrained optimal control problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous state inequality constraints. Each of these constrained optimal parameter selection problems can be regarded as an optimization problem subject to equality constraints and continuous inequality constraints. On this basis, an exact penalty function method is used to devise a computational method to solve these optimization problems with equality constraints and continuous inequality constraints. The main idea is to augment the exact penalty function constructed from the equality constraints and continuous inequality constraints to the objective function, forming a new one. This gives rise to a sequence of unconstrained optimization problems. It is shown that, for sufficiently large penalty parameter value, any local minimizer of the unconstrained optimization problem is a local minimizer of the optimization problem with equality constraints and continuous inequality constraints. The convergent properties of the optimal parameter selection problems with equality constraints and continuous inequality constraints to the original optimal control problem are also discussed. For illustration, three examples are solved showing the effectiveness and applicability of the approach proposed.     

Optimality Conditions in Terms of Alternance: Two Approaches

In Optimization Theory, necessary and sufficient optimality conditions play an essential role. They allow, first of all, checking whether a point under study satisfies the conditions, and, secondly, if it does not, finding a “better” point. For the class of directionally differentiable functions, a necessary condition for an unconstrained minimum requires the directional derivative to be non-negative in all directions. This condition becomes efficient for special classes of directionally differentiable functions. For example, in the case of convex and max-type functions, the necessary condition for a minimum takes the form of inclusion. The problem of verifying this condition is reduced to that of finding the point of some convex and compact set C which is nearest to the origin. If the origin does not belong to C, we easily find the steepest descent direction, and are able to construct a numerical method. In the classical Chebyshev polynomial approximation problem, necessary optimality conditions are expressed in the form of alternation of signs of some values. In the present paper, a generalization of the alternance approach to a general optimization problem is given. Two equivalent forms of the alternance condition (the so-called inside form and the outside one) are discussed in detail. In some cases, it may be more convenient to use the conditions in the form of inclusion, in some other—the condition in the alternance form as in the Chebyshev approximation problem. Numerical methods based on the condition in the form of inclusion usually are “gradient-type” methods, while methods employing the alternance form are often “Newton-type”. It is hoped that in some cases it will be possible to enrich the existing armory of optimization algorithms by a new family of efficient tools. In the paper, we discuss only unconstrained optimization problems in the finite-dimensional setting. In many cases, a constrained optimization problem can be reduced (via Exact Penalization Techniques) to an unconstrained one.     

Optimality conditions for reflecting boundary control problems

We consider a control problem with reflecting boundary and obtain necessary optimality conditions in the form of the maximum Pontryagin principle. To derive these results we transform the constrained problem in an unconstrained one or we use penalization techniques of Morreau-Yosida type to approach the original problem by a sequence of optimal control problems with Lipschitz dynamics. Then nonsmooth analysis theory is used to study the convergence of the penalization in order to obtain optimality conditions.     

Nondegenerate necessary conditions for nonlinear optimal control problems with higher-index state constraints

For some optimal control problems with pathwise state constraints the standard versions of the necessary conditions of optimality are unable to provide useful information to select minimizers. There exist some literature on stronger forms of the maximum principle, the so-called nondegenerate necessary conditions, that can be informative for those problems. These conditions can be applied when certain constraint qualifications are satisfied. However, when the state constraints have higher index (i.e. their first derivative with respect to time does not depend on the control) these nondegenerate necessary conditions cannot be used. This happens because constraint qualifications assumptions are never satisfied for higher index state constraints. We note that control problems with higher index state constraints arise frequently in practice. An example is a common mechanical systems for which there is a constraint on the position (an obstacle in the path, for example) and the control acts as a second derivative of the position (a force or acceleration) which is a typical case. Here, we provide a nondegenerate form of the necessary conditions that can be applied to nonlinear problems with higher index state constraints. When addressing a problem with a state constraint of index k, the result described is applicable under a constraint qualification that involves the k -th derivative of the state constraint, corresponding to the first time when derivative depends explicitly on the control. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Estimation of Exact Penalty for Infinite-Dimensional Inequality-Constrained Minimization Problems

We use the penalty approach in order to study inequality-constrained minimization problems in infinite dimensional spaces. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we consider a large class of inequality-constrained minimization problems for which a constraint is a mapping with values in a normed ordered space. For this class of problems we introduce a new type of penalty functions, establish the exact penalty property and obtain an estimation of the exact penalty. Using this exact penalty property we obtain necessary and sufficient optimality conditions for the constrained minimization problems.     

Pontryagin Type Maximum Principle for budget-constrained infinite horizon optimal control problems with linear dynamics

In this paper a class of infinite horizon optimal control problems with an isoperimetrical constraint, also interpreted as a budget constraint, is considered. Herein a linear both in the state and in the control dynamic is allowed. The problem setting includes a weighted Sobolev space as the state space. For this class of problems, we establish the necessary optimality conditions in form of a Pontryagin Type Maximum Principle including a transversality condition. The proved theoretical result is applied to a linear–quadratic regulator problem.     

General fractional variational problem depending on indefinite integrals

In this report, we consider two kind of general fractional variational problem depending on indefinite integrals include unconstrained problem and isoperimetric problem. These problems can have multiple dependent variables, multiorder fractional derivatives, multiorder integral derivatives and boundary conditions. For both problems, we obtain the Euler-Lagrange type necessary conditions which must be satisfied for the given functional to be extremum. Also, we apply the Rayleigh-Ritz method for solving the unconstrained general fractional variational problem depending on indefinite integrals. By this method, the given problem is reduced to the problem for solving a system of algebraic equations using shifted Legendre polynomials basis functions. An approximate solution for this problem is obtained by solving the system. We discuss the analytic convergence of this method and finally by some examples will be showing the accurately and applicability for this technique.     

Elimination of bounds in optimization problems by transforming variables

The problem of minimizing a nonlinear objective function ofn variables, with continuous first and second partial derivatives, subject to nonnegativity constraints or upper and lower bounds on the variables is studied. The advisability of solving such a constrained optimization problem by making a suitable transformation of its variables in order to change the problem into one of unconstrained minimization is considered. A set of conditions which guarantees that every local minimum of the new unconstrained problem also satisfies the first-order necessary (Kuhn—Tucker) conditions for a local minimum of the original constrained problem is developed. It is shown that there are certain conditions under which the transformed objective function will maintain the convexity of the original objective function in a neighborhood of the solution. A modification of the method of transformations which moves away from extraneous stationary points is introduced and conditions under which the method generates a sequence of points which converges to the solution at a superlinear rate are given.     

New optimality conditions for unconstrained vector equilibrium problem in terms of contingent derivatives in Banach spaces

This article presents necessary and sufficient optimality conditions for weakly efficient solution, Henig efficient solution, globally efficient solution and superefficient solution of vector equilibrium problem without constraints in terms of contingent derivatives in Banach spaces with stable functions. Using the steadiness and stability on a neighborhood of optimal point, necessary optimality conditions for efficient solutions are derived. Under suitable assumptions on generalized convexity, sufficient optimality conditions are established. Without assumptions on generalized convexity, a necessary and sufficient optimality condition for efficient solutions of unconstrained vector equilibrium problem is also given. Many examples to illustrate for the obtained results in the paper are derived as well.     

Exact Solutions in Optimal Design Problems for Stationary Diffusion Equation

We consider two-phase multiple state optimal design problems for stationary diffusion equation. Both phases are taken to be isotropic, and the goal is to find the optimal distribution of materials within domain, with prescribed amounts, that minimizes a weighted sum of energies. In the case of one state equation, it is known that the proper relaxation of the problem via the homogenization theory is equivalent to a simpler relaxed problem, stated only in terms of the local proportion of given materials.

We prove an analogous result for multiple state problems if the number of states is less than the space dimension. In spherically symmetric case, the result holds for arbitrary number of states, and the optimality conditions of a simpler relaxation problem, which are necessary and sufficient, enable us to explicitly calculate the unique solution of proper relaxation for some examples. In contrary to maximization problems, these solutions are not classical.